# 7.6 Modeling with trigonometric equations  (Page 3/14)

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## Describing periodic motion

The hour hand of the large clock on the wall in Union Station measures 24 inches in length. At noon, the tip of the hour hand is 30 inches from the ceiling. At 3 PM, the tip is 54 inches from the ceiling, and at 6 PM, 78 inches. At 9 PM, it is again 54 inches from the ceiling, and at midnight, the tip of the hour hand returns to its original position 30 inches from the ceiling. Let $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ equal the distance from the tip of the hour hand to the ceiling $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ hours after noon. Find the equation that models the motion of the clock and sketch the graph.

Begin by making a table of values as shown in [link] .

$x$ $y$ Points to plot
Noon 30 in $\left(0,30\right)$
3 PM 54 in $\left(3,54\right)$
6 PM 78 in $\left(6,78\right)$
9 PM 54 in $\left(9,54\right)$
Midnight 30 in $\left(12,30\right)$

To model an equation, we first need to find the amplitude.

The clock’s cycle repeats every 12 hours. Thus,

The vertical shift is

There is no horizontal shift, so $\text{\hspace{0.17em}}C=0.\text{\hspace{0.17em}}$ Since the function begins with the minimum value of $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ (as opposed to the maximum value), we will use the cosine function with the negative value for $\text{\hspace{0.17em}}A.\text{\hspace{0.17em}}$ In the form $\text{\hspace{0.17em}}y=A\text{\hspace{0.17em}}\mathrm{cos}\left(Bx±C\right)+D,\text{\hspace{0.17em}}$ the equation is

$y=-24\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi }{6}x\right)+54$

## Determining a model for tides

The height of the tide in a small beach town is measured along a seawall. Water levels oscillate between 7 feet at low tide and 15 feet at high tide. On a particular day, low tide occurred at 6 AM and high tide occurred at noon. Approximately every 12 hours, the cycle repeats. Find an equation to model the water levels.

As the water level varies from 7 ft to 15 ft, we can calculate the amplitude as

The cycle repeats every 12 hours; therefore, $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ is

$\frac{2\pi }{12}=\frac{\pi }{6}$

There is a vertical translation of $\text{\hspace{0.17em}}\frac{\left(15+8\right)}{2}=11.5.\text{\hspace{0.17em}}$ Since the value of the function is at a maximum at $\text{\hspace{0.17em}}t=0,$ we will use the cosine function, with the positive value for $\text{\hspace{0.17em}}A.$

$y=4\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi }{6}\right)\text{\hspace{0.17em}}t+11$

The daily temperature in the month of March in a certain city varies from a low of $\text{\hspace{0.17em}}24\text{°F}\text{\hspace{0.17em}}$ to a high of $\text{\hspace{0.17em}}40\text{°F}\text{.}\text{\hspace{0.17em}}$ Find a sinusoidal function to model daily temperature and sketch the graph. Approximate the time when the temperature reaches the freezing point $\text{\hspace{0.17em}}32\text{°F}\text{.}\text{\hspace{0.17em}}$ Let $\text{\hspace{0.17em}}t=0\text{\hspace{0.17em}}$ correspond to noon.

$y=8\mathrm{sin}\left(\frac{\pi }{12}t\right)+32$
The temperature reaches freezing at noon and at midnight.

## Interpreting the periodic behavior equation

The average person’s blood pressure is modeled by the function $\text{\hspace{0.17em}}f\left(t\right)=20\text{\hspace{0.17em}}\mathrm{sin}\left(160\pi t\right)+100,\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}f\left(t\right)\text{\hspace{0.17em}}$ represents the blood pressure at time $\text{\hspace{0.17em}}t,$ measured in minutes. Interpret the function in terms of period and frequency. Sketch the graph and find the blood pressure reading.

The period is given by

In a blood pressure function, frequency represents the number of heart beats per minute. Frequency is the reciprocal of period and is given by

$\begin{array}{c}\frac{\omega }{2\pi }=\frac{160\pi }{2\pi }\\ =80\end{array}$

See the graph in [link] .

## Modeling harmonic motion functions

Harmonic motion is a form of periodic motion, but there are factors to consider that differentiate the two types. While general periodic motion applications cycle through their periods with no outside interference, harmonic motion requires a restoring force. Examples of harmonic motion include springs, gravitational force, and magnetic force.

can you not take the square root of a negative number
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas